Abstract: We give a quadratic algorithm for finding the minimum number of reversals needed to sort a signed permutation. Our algorithm is faster than the previous algorithm of Hannenhalli and Pevzner and its faster implementation of Berman and Hannenhalli. The algorithm is conceptually simple and does not require special data structures. Our study also considerably simplifies the combinatorial structures used by the analysis. 2 refs.

Abstract: A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,BDHW,BFH, DEF,DF1-7,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in $k$ steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word $x$ can be derived in a grammar $G$ in $k$ steps, or whether a permutation has a factorization of length $k$ over a given set of generators. We show hardness and completeness for these problems for various levels of the $W$ hierarchy. In particular, we show that Short TM Computation is complete for $W[1]$ . This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes.

Abstract: We investigate two methods for proving lower bounds on the size of small-depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that our main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An $o(n)$-bit protocol for a communication game for computing shifts, which also gives an upper bound of $o(n^2)$ on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives an $o(n)$ upper bound for computing all permutations and an $O(n\log\log n)$ upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic-depth circuits; this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model. (iii) An upper bound on Boolean circuits of depth 2 for computing shifts and, more generally, all permutations; this shows that such circuits are more efficient than the model based on sending bits along vertex-disjoint paths.

Abstract: In recent years, researchers have invested a lot of effort in trying to design suitable alternatives to the RSA signature scheme, with lower computational requirements. The idea of using polynomial equations of low degree in several unknowns, with some hidden trap-door, has been particularly attractive. One of the most noticeable attempts to push this idea forward is the Ong-Schnorr-Shamir signature scheme, which has been broken by Pollard and Schnorr. At Crypto '93, Shamir, proposed a family of cryptographic signature schemes based on a new method. His design made subtle use of birational permutations over the set ofk-tuples of integers modulo a large numberN of unknown factorization. However, the schemes presented in Shamir's paper are weak. In the present paper, we describe several attacks which can be applied to schemes in this general family.

Abstract: In this paper, we present several new and generalized parallel dense matrix multiplication algorithms of the form C = αAB + β C on two-dimensional process grid topologies These algorithms can deal with rectangular matrices distributed on rectangular grids We classify these algorithms coherently into three categories according to the communication primitives used and thus we offer a taxonomy for this family of related algorithms All these algorithms are represented in the data distribution independent approach and thus do not require a specific data distribution for correctness The algorithmic compatibility condition result shown here ensures the correctness of the matrix multiplication We define and extend the data distribution functions and introduce permutation compatibility and algorithmic compatibility We also discuss a permutation compatible data distribution (modified virtual 2D data distribution) We conclude that no single algorithm always achieves the best performance on different matrix and grid shapes A practical approach to resolve this dilemma is to use poly-algorithms We analyze the characteristics of each of these matrix multiplication algorithms and provide initial heuristics for using the poly-algorithm All these matrix multiplication algorithms have been tested on the IBM SP2 system The experimental results are presented in order to demonstrate their relative performance characteristics, motivating the combined value of the taxonomy and new algorithms introduced here © 1997 by John Wiley & Sons, Ltd

Abstract: We study permutation type solutions to n-simplex equations, that is, solutions whose R matrix can be written as a product of delta- functions depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of the n-simplex equation reduce to an [n(n+1)/2+1]x[n(n+1)/2+1] matrix equation over Z_D. We have completely analyzed the 2-, 3- and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.

Abstract: The robust sib-pair method introduced by Haseman & Elston (1972) is one of the most widely circulated allele-sharing methods for linkage analysis. The procedure evaluates linkage by significance testing of a regression coefficient and, hence, a standard t-test has traditionally been applied despite known violations of the statistical assumptions underlying the test. We present a permutation based reference distribution for the estimate of the regression coefficient that is motivated by genetic principles rather than by standard regression testing procedures. The permutation test approximates Mendelian co-segregation under the null hypothesis of no linkage, making it a very natural approach. Theory and simulations show that the conventional t-test approximates the permutation test quite well, even when dependent sib pairs are used for analysis. These results thus indirectly address concerns over the t-test. To illustrate the permutation test using real data we applied the procedure to two lipoprotein systems that have been well characterized.

Abstract: A method and apparatus for inter-round mixing in iterated block substitution systems is disclosed. The method involves optimizing inter-round mixing so that each data bit affects each other data bit in the same way. This is accomplished by applying a quick trickle permutation or a quasi quick trickle permutation to the data bits undergoing block substitution allocated to n individual substitution boxes.

Abstract: Distance functions permeate the field of genetic algorithms especially in relation to mating strategies, incest prevention, diversity preservation, and techniques that find multiple solutions. Distance functions can be found in the literature for genotypes which use binary or numerical parameter encodings. Various phenotypic distance functions that act on the properties of the decoded genotype have also been presented. There is, however, a gap in the literature regarding distance functions for order-based encodings. This paper presents a new distance function based on allele adjacency for permutation encodings. It is shown that this distance function is one possible member of a family of order-based metrics that respond to different properties in order-based encodings. This, and other, new distance measures will allow a wider range of genetic algorithm techniques to be applied to problems in the order-based domain.

Abstract: In this paper, we deal with the famous job-shop scheduling problem, which has been being a constant subject of study for many years due to its high computational complexity (NP-hard in the strong sense). We present a permutation-based scheme for solving the problem, which in the abstraction level differs from the classical one of Jacques Carlier and Eric Pinson. In particular, we specify the differences both in the fashion of stating the constraints (the use of the generalized sorting constraint) and in the search strategy (splitting intervals of task orders). We will first give a constraint program for solving the problem, which involves only primitive constraints and which is clean and simple to understand. We then study some special techniques based on testing variable bounds that allow us to solve two hard instances la21 and la38. These two instances have been open problems recommended in a paper of David Applegate and William Cook in 1991.

Abstract: The Block Sorting Lossless Data Compression Algorithm (BWT) described by Burrows and Wheeler has received considerable attention. Its compression rates are simliar to context-based methods, such as PPM, but at execution speeds closer to Ziv-Lempel techniques. This paper describes the Lexical Permutation Sorting Algorithm (LPSA), its theoretical basis and delineates its relationship to BWT. In particular we describe how BWT can be reduced to LPSA and show how LPSA could give better results than BWT when transmitting permutations.

Abstract: We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.

Abstract: A new procedure to analyze the results of experimental designs is introduced. A permutation procedure based on Euclidean distance is used to evaluate residuals obtained from least sum of absolute deviations regression. Applications include completely randomized and randomized block configurations including one-way, factorial, split-plot, and Latin square designs, with or without covariates. Parametric assumptions of homogeneity of variance, compound symmetry, and normality are eliminated with this procedure.

Abstract: We introduce the notion of double permutation in order to study particular classes of transformations of the one-dimensional cellular automata rule space. These classes of transformations are characterized according to different sets of metrical, language theoretic, and dynamical properties they preserve. Each set of transformations we propose induces an equivalence relation over the cellular automata rule space. We give exact results on the cardinality of the quotient sets generated by these equivalence relations. Finally, we discuss some interesting open problems.

Abstract: In this note, we show that the discrepancy of any family of ‘ permutations of [n] = f1; 2;:::;ng is O( p ‘ logn), improving on the O(‘ logn) bound due to Bohus (Random Structures & Algorithms, 1:215{220, 1990). In the case where ‘ n, we show that the discrepancy is (min f p n log(2‘=n);ng).

Abstract: Boorges, in the essay "Kafka and His Precursors," suggests that our perception of the present alters our conception of the past, that we can look at texts from the past in a new way, influenced by things we now understand.3 By the light of the computer, then, we can look anew at a long history of mystical texts and combinatorial systems that reach back to antiquity. Mystical systems involving permutational procedures that purport to reveal a body of hermetic knowledge or that lead to a revelatory exhaustion of all possibilities prefigure the computer's potential to permute and, given rules, to engage in "creative magic" by finding meaning in new combinations. A number of artists in this century, with or without the computer, have explored this realm in their work. I have searched for examples of three specific types of combinatorial systems. In mathematics, these three types are called permutation, combination, and variation. Each begins with a limited number of items, a set of things. In permutations, the positions of these things are shuffled within the whole set, as in an anagram. For combinations, one can take out any number of elements from the set and put them together in a smaller group. Variations are permutations with repetitions allowed; in variations, one can permute to infinity. The computer, of course, excels at all of these systematic activities. During this search for examples, certain questions, themes, and comparisons arose. Why are permutations of abstract symbols so often linked to creation, whether divine or artistic? What is it about permuting letters or numbers that leads to mystical experience? Is this experience born out of the creative transformation that occurs or out of the meditative activity? What role can the computer play as a stand-in for this process? What is the qualitative difference between permutational systems that are intentionally driven, and those systems that are manipulated with chance operations? Among the themes that have recurred is the notion of total exhaustion; it is often hinted that, if all the possibilities of permutations are exhausted, there might be a revelation or a transformation on a larger scale, or even the end of the world. There is the recurring idea that the numbers 1 or 2 or 3 can give birth to everything there is, to the infinite. The number 1 is often paradoxically equated with the infinite. Comparing the historic mystical systems to twentieth-century artistic practice with a similar systematic basis presents other questions. Are there larger, more transfor-

Abstract: Sorting by tranpositions is the problem of finding the minimum number of transpositions required to sort a permutation pi A transposition involves repositioning a contiguous sequence (block) of elements by inserting it elsewhere in the permutation The problem has applications in the study of genome rearrangements and phylogeny reconstruction In this paper, several heuristics based on analyses of subsequences and runs in a permutation are employed Experimental results are provided The algorithm based on the longest increasing subsequence in a permutation appears most promising

Abstract: Let Λ be an arbitrary set of positive integers andSn(Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |Sn(Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |Sn(Λ)| stands for the number of elements in the finite setSn(Λ).

Abstract: In this correspondence, we first propose a new two-dimensional (2-D) Hadamard transform algorithm, which can be realized in fixed and identical pipeline stages. By introducing exchangeable permutations, the fixed-pipeline algorithm can be further extended to provide all 2-D lower-dimension transformations in intermediate pipeline stages. Finally, the parallel pipeline realization of the proposed algorithm is also suggested. For VLSI implementation, the proposed fixed-pipeline structure with the same computational complexity provides better modularity than the other famous algorithms. With lower dimension transformations, the proposed algorithm is suitable for applications in variable-block-size compression.

Abstract: A method and apparatus for inter-round mixing in iterated block substitution systems is disclosed. The method involves optimizing inter-round mixing so that each sub-block of data affects each other in the same way. This is accomplished by applying a quick trickle permutation or a quasi quick trickle permutation to blocks of data allocated to n individual substitution boxes.